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Inverse multiobjective optimization provides a general framework for the unsupervised learning task of inferring parameters of a multiobjective decision making problem (DMP), based on a set of observed decisions from the human expert. However, the performance of this framework relies critically on the availability of an accurate DMP, sufficient decisions of high quality, and a parameter space that contains enough information about the DMP. To hedge against the uncertainties in the hypothetical DMP, the data, and the parameter space, we investigate in this paper the distributionally robust approach for inverse multiobjective optimization. Specifically, we leverage the Wasserstein metric to construct a ball centered at the empirical distribution of these decisions. We then formulate a Wasserstein distributionally robust inverse multiobjective optimization problem (WRO-IMOP) that minimizes a worst-case expected loss function, where the worst case is taken over all distributions in the Wasserstein ball. We show that the excess risk of the WRO-IMOP estimator has a sub-linear convergence rate. Furthermore, we propose the semi-infinite reformulations of the WRO-IMOP and develop a cutting-plane algorithm that converges to an approximate solution in finite iterations. Finally, we demonstrate the effectiveness of our method on both a synthetic multiobjective quadratic program and a real world portfolio optimization problem. 

Inverse multiobjective optimization provides a general framework for the unsupervised learning task of inferring parameters of a multiobjective decision making problem (DMP), based on a set of observed decisions from the human expert. However, the performance of this framework relies critically on the availability of an accurate DMP, sufficient decisions of high quality, and a parameter space that contains enough information about the DMP. To hedge against the uncertainties in the hypothetical DMP, the data, and the parameter space, we investigate in this paper the distributionally robust approach for inverse multiobjective optimization. Specifically, we leverage the Wasserstein metric to construct a ball centered at the empirical distribution of these decisions. We then formulate a Wasserstein distributionally robust inverse multiobjective optimization problem (WRO-IMOP) that minimizes a worst-case expected loss function, where the worst case is taken over all distributions in the Wasserstein ball. We show that the excess risk of the WRO-IMOP estimator has a sub-linear convergence rate. Furthermore, we propose the semi-infinite reformulations of the WRO-IMOP and develop a cutting-plane algorithm that converges to an approximate solution in finite iterations. Finally, we demonstrate the effectiveness of our method on both a synthetic multiobjective quadratic program and a real world portfolio optimization problem. 

Abstract We study a class of bilevel spanning tree (BST) problems that involve two independent decision‐makers (DMs), the leader and the follower with different objectives, who jointly construct a spanning tree in a graph. The leader, who acts first, selects an initial subset of edges that do not contain a cycle, from the set under her control. The follower then selects the remaining edges to complete the construction of a spanning tree, but optimizes his own objective function. If there exist multiple optimal solutions for the follower that result in different objective function values for the leader, then the follower may choose either the one that is the most (optimistic version) or least (pessimistic version) favorable to the leader. We study BST problems with the sum‐ and bottleneck‐type objective functions for the DMs under both the optimistic and pessimistic settings. The polynomial‐time algorithms are then proposed in both optimistic and pessimistic settings for BST problems in which at least one of the DMs has the bottleneck‐type objective function. For BST problem with the sum‐type objective functions for both the leader and the follower, we provide an equivalent single‐level linear mixed‐integer programming formulation. A computational study is then presented to explore the efficacy of our reformulation.